A Resolution-based Model Building Algorithm for a Fragment of $\text{OCC1N}$₌

Abstract

$\text{OCC}$1 $\text{N}$ [11] is a decidable subclass of first-order clausal logic without equality. [7] shows that $\text{OCC}$1 $\text{N}$ becomes undecidable when equational literals are allowed, but remains decidable if equality is restricted to ground terms only.

First, we extend this decidability result to some non ground equational literals. By carefully restricting the use of the equality predicate we obtain a new decidable class, called $\text{OCC}$1 $\text{N}$₌^∗. We show that existing paramodulation calculi do not terminate on $\text{OCC}$1 $\text{N}$₌^∗ and we define a new simplification rule which allows to ensure termination. Second, we show that the automatic extraction of Herbrand models is possible from saturated sets in $\text{OCC}$1 $\text{N}$₌^∗ not containing □. These models are represented by certain finite sets of (possibly equational and non ground) linear atoms. The difficult point here is to show that this formalism is suitable as a model representation mechanism, i.e. that the evaluation of arbitrary non equational first-order formulae in such interpretations is a decidable problem.